Curved grating spectrometer with very high wavelength resolution

ABSTRACT

The present application discloses a system comprising a compact curved grating (CCG) and its associated compact curved grating spectrometer (CCGS) or compact curved grating wavelength multiplexer/demultiplexer (WMDM) module and a method for making the same. The system is capable of achieving a very small (resolution vs. size) RS factor. In the invention, the location of the entrance slit and detector can be adjusted in order to have the best performance for a particular design goal. The initial groove spacing is calculated using a prescribed formula dependent on operation wavelength. The location of the grooves is calculated based on two conditions. The first one being that the path-difference between adjacent grooves should be an integral multiple of the wavelength in the medium to achieve aberration-free grating focusing at the detector or output slit (or output waveguide) even with large beam diffraction angle from the entrance slit or input slit (or input waveguide). The second one being specific for a particular design goal of a curved-grating spectrometer. In an embodiment, elliptical mirrors each with focal points at the slit and detector are used for each groove to obtain aberration-free curved mirrors.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation-In-Part of U.S. patent applicationSer. No. 10/708,730, filed Mar. 20, 2004 now U.S. Pat. No. 7,283,233.This application is related to U.S. patent application Ser. No.09/916,701, “Method for shifting bandgap energy of quantum well layer”filed Jul. 26, 2001 in the names of Boon-Siew Ooi and Seng-Tiong Ho, thedisclosure of which is incorporated herein by reference. Thisapplication is related to U.S. Patent Application Ser. No. 60/242,219,“Method for shifting bandgap energy of quantum well layer” filed Oct.20, 2000, in the names of Boon-Siew Ooi and Seng-Tiong Ho, thedisclosure of which is incorporated herein by reference. Thisapplication is related to U.S. Patent Application Ser. No. 60/430,507,“Method for quantum-well intermixing using pre-annealing enhanced defectdiffusion” filed Dec. 3, 2002, in the names of Boon-Siew Ooi and Ruiyu(Jane) Wang, the disclosure of which is incorporated herein byreference.

BACKGROUND OF THE INVENTION

The present invention relates to semiconductor photonic andopto-electronic devices. In particular, the present invention relates toan integrated optical wavelength multiplexers and demultiplexers andmethod of making the same.

Optical gratings are well known in the art and are used to disperseoptical spectra spatially. Such gratings are commonly used in opticalspectrometers to analyze the spectra composition of an optical beam.There is always a trade off between the length of an opticalspectrometer and its resolution. Thus, if a higher wavelength resolutionis required, the length required is also longer. Consider an example ofa typical 1-meter long grating spectrometer in the market, which has awavelength resolution of about Δλ=0.1 nm at λ=1000 nm or Δλ/λ=10⁻⁴. Thedimensionless quantity for the length of the spectrometer L is L/λ andL/λ=10⁶ in this example. The dimensionless product of the relativeresolution Δλ/λ and the relative physical size L/λ of the spectrometeris dependent on the design of the spectrometer and in this example,spectrometer gives (Δλ/λ)×(L/λ)=100=RS. This factor (RS) in generallyreferred to as the “resolution vs size” factor. RS basically measuresthe compactness of a spectrometer for a given resolution power. Thesmaller the RS value, the more compact is the spectrometer. Only a fewconventional spectrometers have RS factor less than about 10. This isprimarily because of the various limitations in the current art (as willbe described below).

It is known in the art that a relatively compact spectrometer can beachieved using a curved grating. The schematics of such a gratingspectrometer is shown in FIG. 1A, illustrating an optical beam 1entering an entrance slit 2 with slit size w₁. The beam, after slit 2,undergoes wave diffraction towards a curved grating 3, which diffractsthe beam spatially in a direction that is dependent on the opticalwavelength of the beam. The curvature of the grating helps to refocusthe diffracted beam to an exit slit 4 with slit size w₂. Light throughslit 4 is then detected by a photo detector 5. As is well known to thoseskilled in the art, the commonly used design for the curved grating 3 isthe Rowland design. In the Rowland design, the grating has a circularlycurved shape of radius R 6 and the slits SL1 and SL2 lie in a circle ofradius R/2 7 as shown in FIG. 1A. The grating is ruled using a diamondtip with constant horizontal displacement d, which ruled the curvedsurface with constant Chord lines C1, C2, C3 and so forth, as shown inFIG. 1B. The segment lengths, S1, S2, S3 and so forth, along the curvedsurface are not a constant and vary along the curved surface.

Let the diffraction full angle from the entrance slit 1 be θ_(div). Asis well known to those skilled in the art, θ_(div)=2λ/d (in Radian). Letlength L be the distance between the grating center and entrance slit 1,which is also approximately the distance between the grating center andthe exit slit 4. as is well know to those skilled in the art, theresolution of the spectrometer increases with decreasing slit size w₂.The imaging through the curved grating requires w₁ and w₂ to be aboutequal. a smaller slit size w₁ and w₂ to be about equal. A smaller slitsize w₁, however, leads to a larger diffraction angle θ_(div). It can beshown that the Rowland design works reasonably well up to θ_(div)<4°.When θ_(div)>4°, the Rowland design could not give a sharp enough focusat the exit slit 4 (for Δλ<0.1 nm), thereby limiting the size of w₂ andhence the resolution of the spectrometer. A diffraction angle ofθ_(div)=4° corresponds to a slit size of about 25 microns (for λ=1000nm). In the current art, it is typically difficult to make slit sizesmaller than 25 microns, and Rowland design is adequate for most presentspectrometers with slit sizes larger than 25 microns.

Aberration limitation. In the case of the Rowland design, when Θ_(div)>4DEG, serous aberration in the refocusing beam will occur to limitwavelength resolution. This is shown in FIG. 1C illustrating the raytracing for the typical Rolwand-Echelle design at 4, 8, and 16 DEGdiffraction. The ray tracing will allow us to see potential focusingdistortion or aberration at the exit slit. In the figure, we show thefocusing behavior for two sets of rays with wavelengths separated by 0.4nm. From the figure, we see that their focused spots clearly separatewhen Θ_(div)=4 DEG. However, when Θ_(div)=8 DEG, the focused spots beganto smear out. There is substantial distortion for the focusing rays whenΘ_(div)>4 DEG. Further simulations based on numerical solutions toMaxwell's wave equations using finite-difference time-domain (FDTD)method also show similar onset of focused spot size distortion atΘ_(div)>4 DEG. In short, the current designs are close to theirresolution-size (RS) limits and cannot be made substantially morecompact without losing wavelength resolution.

As discussed above, a curved-grating spectrometer is well specified bygeometric configurations of its components as shown in FIG. 2. First,the location of the entrance slit; this is usually given by an θ₁ withrespect to the normal of the grating center and the distance S₁ from thegrating center. The center of the grating refers to the part of thegrating hit by the center, i.e. high intensity point, of the entrancebeam. Second, the locations of the first two grooves at the gratingcenter; these are specified by its location vectors X₁ and X⁽⁻¹⁾ withrespect to the grating center X₀=0 and its groove spacing (or pitch)d₁=|X₁−X₀| and d⁻¹=|X⁽⁻¹⁾−X₀|. X₁ and X⁽⁻¹⁾ are located symmetricallyopposite to each other with respect to the grating center and therefored=d₁=d⁽⁻¹⁾. A circle can be defined by these three points X₀, X₁, and X₂and its radius is referred to as the radius is referred to as the radiusof curvature at the grating center. Third, the location of the exit sliti.e. the location of the detector; this is specified by an angle θ₂ withrespect to the normal of the grating center and the distance S₂ from thegrating center. For a given operating wavelength center λ_(c), theinitial groove spacing d is usually chosen to satisfy the diffractiongrating formula for a given entrance slit and detector location. Thecurved grating is further specified by the location of other grooves(specified by its location vector X_(i), with respect to the gratingcenter X₀=0 and the groove spacing d_(i) from the previous groove givenby d_(i)=|X_(i)X_(i−1)|. Let the total number of groove be N in eachside of the grating center, the locus of all the groove X_((−N)), . . ., X⁽⁻¹⁾, X₀, X₁, . . . , X_(N) form a curved shape, which can lie in acircle or in any other curvilinear line. Curved shape of the gratingacts as an imaging element of the spectrometer.

The shape of each groove centered at X_(i) is not critical to theresolution power of the grating and hence is not necessary to be a partof the main specification. However, the groove shape is related to thediffraction efficiency. For example, in order to increase thediffraction efficiency at a particular diffraction angle θ₂, it istypically made a planar surface for each groove, oriented in such a waythat it acts like a tiny mirror reflecting the input ray towards theangle θ₂, a process typically referred to as blazing to angle θ₂ (for agiven wavelength λ). A section of each groove which reflects light isphysically a two-dimensional surface of a particular shape, not aone-dimensional curve. However, the geometric shape of a groove isusually referred to as a curve of a particular shape. This is becausethere is no variation in the grating shape in the directionperpendicular to the plane where grating lies. Especially, spectrometerswithin a planar waveguide are strictly two-dimensional in their natureand the shape of grating or grooves will be referred with a curve, notwith surface.

Conventional Rowland design spectrometers are specifically configured bythe design rule described below in conjunction with FIG. 3.

Referring to FIG. 3, the entrance slit is located on a circle of R/2,where R is the radius of curvature at the grating center. This circle ofradius R/2 is called as Rowland circle and it is tangent to the gratingcenter. In the Rowland design, the distance S₁ of the entrance slit tothe grating center is related to the angle of incidence θ₁ byS₁=R×cosθ₁.

The detector is also located on the same Rowland circle as the entranceslit. In the Rowland design, the distance S₂ of the detector to thegrating center is related to the angle of diffraction θ₂ by S₂=R×cosθ₂.

The relation between θ₁, θ₂, and the initial groove spacing d is givenby the grating formula,d(sinΘ₂−sinΘ₁)=mλ _(c) /n  (1)

where m is the diffraction order, n is the refractive index of themedium, λ_(c) is the center of the operation wave-length. This gratingformula is a so-called far-field approximation, which is valid only whenS₁ and S₂ are much larger than d.

Initial groove positions are X₀=(0,0), X₁=(d, R−(R²−d²)^(1/2)) andX⁻¹=(−d, R−(R²−d²)^(1/2)). These three initial grooves with positionvectors X₀, X₁, and X⁻¹, are located on a circle of radius R and havethe initial groove spacing of d along a chord parallel to the gratingtangent.

All other grooves, specified by its position vector X₁'s, are located onthe same circle of radius R defined by the initial three groovepositions X, X, and X⁻¹. X₁'s are also equally spaced along a chord thatis parallel to the tangent of the grating center. In other words, theprojection of the displacement vector X₁−X_(i−1) on this chord alwayshas the same length. Specifically, the position vectors of these groovescan be written as x_(i)=(di, R−(R²−(di)²)^(1/2)), and X_(−i)=(di,R−(R²−(di)²)^(1/2).

For example, if the radius of curvature at the grating center is r=100μm, the Rowland circle, where the entrance slit and the detector arelocated, has the radius of 50 μm. Here, we assume that tangent line atthe grating center is parallel to the x-axis. Since the Rowland circleis tangent to the grating center, it circles by passing both the gratingcenter X₀=(0,0) and a point (0, 50). If the angle of the entrance slitis θ₁=45°, the distance of the entrance slit to the grating center isS₁=R×cosθ₁=70.71 μm. In terms of (x,y)-coordinate, the entrance slit islocated at (−50, 50). It is well-known that grating is more efficient ifthe propagation direction of the diffracted light from the grating isparallel and opposite to the propagation direction of the input beam.Such a scheme is known as Littrow configuration and is widely used for ahigh-efficiency spectrometer. A Littrow configuration in the Rowlanddesign will be equivalent to having the angle of detector being almostequal to the angle of the entrance slit, i.e., θ₁≈θ₂. In order to haveLittrow configuration, the groove spacing d at the grating center has tobe properly chosen so that it satisfies grating formula Eq. 1. Forexample, when the center wavelength is 1550 nm and the angle of entranceslit is θ₁=45°, the diffraction order of m=12 of a grating with thegroove spacing of d=4.2 μm at its center propagate toward a detectorlocated at θ₂=37.37°, which is close to the Littrow configuration. Thedetector location can be fine tuned by changing the initial groovespacing d. Lower the groove spacing d, larger the detector angle θ₂. Forthe groove spacing d=4.2 μm and radius of curvature R=100 μm, theinitial three positions of grooves are X₀=(0,0), X₁=(4.2, 0.088) andX⁻¹=(−4.2, 0.088).

In the Rowland design, other grooves are located such that their spacingis the same along a chord parallel to the grating tangent at the center.Therefore, the position vectors of other grooves are X_(i)=(di,R−(R²−(di)²)^(1/2))=(4.2i, 100−(100²−(4.2i)²)^(1/2)), and X_(−i)=(−di,R−(R²−(di)²)^(1/2))=(−4.2i, 100−(100²−(4.2i)²)^(1/2)). The positionvectors of the grooves are listed in the following table for the case ofRowland design with R=100 μm, d=4.2 μm, m=12, θ₁=45°, and θ₂=37.37° foran operation wavelength of λ_(c)=1550 nm.

TABLE 1 X⁻¹³ (−54.6, 16.221) X⁻¹² (−50.4, 13.630) X⁻¹¹ (−46.2, 11.312)X⁻¹⁰ (−42, 9.248) X⁻⁹ (−37.8, 7.419) X⁻⁸ (−33.6, 5.814) X⁻⁷ (−29.4,4.419) X⁻⁶ (−25.2, 3.227) X⁻⁵ (−21, 2.230) X⁻⁴ (−16.8, 1.421) X⁻³(−12.6, 0.797) X⁻² (−8.4, 0.353) X⁻¹ (−4.2, 0.088) X₀ (0, 0) X₁ (4.2,0.088) X₂ (8.4, 0.353) X₃ (12.6, 0.797) X₄ (16.8, 1.421) X₅ (21, 2.230)X₆ (25.2, 3.227)

The advent in Dense Wavelength Division Multiplexing (DWDM) opticalcommunication networks, however, requires that the multiple wavelengthsin an optical fiber to be analyzed by spectral analysis devices that aremuch smaller in size than that of the current grating spectrometer. Thechallenge is to circumvent the current limitation in gratingspectrometer design and fabrication methods. As discussed above, thecurrent design basically cannot achieve the Resolution-Size factor (RS)much smaller than about 10. While several current technologies arecapable of using planar waveguide technologies to make grating basedspectrometers on a single silica or semiconductor substrate, they arestill not able to achieve RS much smaller than 10 due to the basiclimitations of the grating spectrometer design. Achieving a smaller RSfactor is important for combining or integrating high-resolution gratingspectrometers with various photonic devices (such as lasers, modulators,or detectors in a compact module or silica/silicon/semiconductor wafer).

These wave-length-division-multiplexed (WDM) integrated photonic devicesor modules would be of great importance for applications to DWDMnetworks. The costs of these integrated WDM devices are typicallyproportional to their sizes. The wavelength dispersion elements, such asthe grating spectrometer or other form of wavelength filters, aretypically about 100 times larger in size than any other photonic devicesin the module or wafer. In order to reduce their costs substantially, itis desirable to reduce the size of these wavelength dispersion elementsto as small a size as possible.

Thus, it is desirable to have grating based spectrometers that have anRS factor of less than 10. It is also desirable to reduce the size, andhence the cost, of integrated WDM devices that are used in DWDMnetworks. The present invention discloses such a device and a method formaking the same.

SUMMARY OF THE INVENTION

It is an aim of the invention to provide a compact curved grating andassociated compact curved grating spectrometer or wavelength Mux/deMuxwith integration possibility that is capable of achieving very small RSfactors thereby enabling high resolution at small size.

It is another aim of the invention to provide a compact curved gratingspectrometer module that can be used as an isolated optical spectrometerusing discrete optical spectrometer module or wavelength Mux/deMuxmodule with integration possibility that can be used as an isolatedoptical spectrometer using discrete optical components.

It is another aim of the invention to provide a compact curved gratingspectrometer module that can be used as a wavelength dispersion elementin a photonic integrated circuit.

In order to attain the above-mentioned aims, a compact curved gratingand associated compact curved grating spectrometer or wavelengthMux/deMux with integration possibility is provided. The compact curvedgrating spectrometer includes an entrance slit, a detector and a curvedgrating and the compact curved grating wavelength Mux/deMux withintegration possibility includes at least an input slit or waveguide andat least an output slit or waveguide for propagating through at least aninput light beam and at least an output light beam, respectively. Thelocations of the entrance slit and the detector or the input slit (orwaveguide) and the output slit (or waveguide) can be adjusted to controlthe performance of the spectrometer or wavelength Mux/deMux. Thedistance between the grooves of the gratings depend on the location ofthe entrance slit or the input slit (or waveguide), the detector or theoutput slit (or waveguide), the center of the operation wavelength, thediffraction order and the refractive index of the medium.

BRIEF DESCRIPTION OF THE DRAWINGS

The preferred embodiments of the invention will hereinafter be describedin conjunction with the appended drawings provided to illustrate and notto limit the invention, wherein like designations denote like elements,and in which FIG. 1A and FIG. 1B show different views of a curvedgrating having the Rowland design, and FIG. 1C shows ray-tracing for aRowland grating design indicating focusing distortion or aberration atthe exit slit for the cases where the input divergence angles are 4 DEG(left), 8 DEG (middle), and 16 DEG (right); FIG. 2 shows the generalspecification of a curved-grating spectrometer; FIG. 3 illustrates theRowland configuration specification for a Rowland curved grating; FIG. 4shows a specific case of Rowland design curved grating; FIG. 5Adescribes the Comparison of angular resolution for Rowland gratin (left)with input divergence angle 16DEG and the HR-CCG design with large-angleaberration correction at input divergence angle 50DEG (right); FIG. 5Bdescribes High Resolution Compact Curved Gratin specifications inaccordance with a preferred embodiment of the present invention; FIG. 6illustrates an example of High Resolution Compact Curved Grating withconstant angle; FIG. 7 shows a High Resolution Compact Curved Gratingwith Constant Arc, the detector and the entrance slit being present on atangent circle, in accordance with an embodiment of the presentinvention; and FIGS. 8A and 8B show a comparison between the Rowlanddesign and the High Resolution Compact Curved Grating with constantgroove (arc) length in accordance with the present invention.

FIG. 9 illustrates a block diagram of a wavelength demultiplexor and awavelength multiplexor.

FIG. 10 illustrates a curved grating spectrometer configured as awavelength multiplexor.

FIG. 11 illustrates a curved grating spectrometer configured as awavelength multiplexor.

FIG. 12 illustrates a curved grating spectrometer configured as awavelength multiplexor/demultiplexor.

FIG. 13 illustrates a wavelength spectrometer configured as amultiplexor/demultiplexor with waveguides.

FIG. 14 illustrates a high resolution compact curved grating.

DETAILED DESCRIPTION

The present invention discloses a system comprising a compact curvedgrating (CCG), it associated compact curved grating spectrometer (CCGS)or wavelength Mux/deMux (WMDM) module and a method for making the same.The system is capable of achieving very small (resolution vs. size) RSfactor. The uses of CCGS or WMDM module include an isolated opticalspectrometer or wavelength Mux/deMux using discrete optical componentssuch as slits, grating, spectrometer or wavelength Mux/deMux casing,detector, detector array, and motor drive. More generally, the CCGS orWMDM module could also be used as a wavelength dispersion element in aphotonic integrated circuit. The photonic integrated circuit can bebased on either of glass (silica) waveguide, semiconductor waveguide(including but not limited to, polymer waveguide, or any other type ofoptical waveguiding devices. Semiconductor waveguides include silicon orcompound semiconductor waveguides such as III-V (GaAs, InP etc). Thewavelength dispersion element based on the CCGS or WMDM module in thephotonic integrated circuit can be integrated with optical detector,laser, amplifier, modulator, splitter, multimode interference devices,other wavelength filters, array-waveguide-based devices, and otherphotonic devices, materials, or components to achieve a multi-componentphotonic integrated circuit with useful functionalities. The CCGexplained below is a High Resolution Compact Curved Grating (HR-CCG)that tries to alleviate the disadvantages associated with prior artmentioned earlier, by providing a high resolution in a small (compact)module.

We have improved on the current Rowland design, enabling curved-gratingspectrometer with 10-100× smaller linear size (or 100-10,000× smallerarea) using our HR-CCG with large-angle aberration-corrected design. Thetypical Rowland design can only reach a useful diffraction angleΘ_(diff) of ˜4 DEG, beyond which serous aberration in the refocusingbeam will occur to limit wavelength resolution. In FIG. 5A we show theangular resolution of the typical Rowland design at 16 DEG diffractionangle compared with our HR-CCG design at 50 DEG. We see that our“large-angle aberration-corrected grating” design has much betterangular resolution: different direction rays are well converged to apoint on the focal circle. This translates to much smaller RS factor orsize. We have used discrete time solution of vectorial Maxwell'sequations to simulate the HR-CCG design, which verified the highresolution nature of our grating as predicted by the ray-tracing method.

Referring to FIG. 9, a wavelength demultiplexer (wavelength deMux) 100is a device in which multiple wavelengths in a beam of light 112 areseparated into several different beams of light 114. A wavelengthmultiplexer (wavelength Mux) 110 is a device in which multiplewavelengths in several beams of light 114 are combined to a single beamof light 112.

Referring to FIG. 9, a wavelength demultiplexer (wavelength deMux) 100is a device in which multiple wavelengths in a beam of light 112 areseparated into several different beams of light 114. A wavelengthmultiplexer (wavelength Mux) 110 is a device in which multiplewavelengths in several beams of light 114 are combined to a single beamof light 112.

Referring to FIG. 10 a curved grating spectrometer functions as awavelength deMux 100 if it has an input slit 116 at the input beam 112location so that the multiple wavelengths in the input beam will bediffracted to several output points 114 and several output slits 118 areplaced at the locations of the spectrometer detectors mentioned above atthese output points to form several output beams each with a differentwavelength. The output slits 118 replace the detectors so that nodetectors will be used.

Referring to FIG. 11, a curved grating spectrometer functions as awavelength Mux 110 if it has multiple input slits 116, each slit placedat an input beam's 114 location so that all the input beams 114 will bediffracted to the same output point and an output slit 118 is placed atthe location of the spectrometer detector mentioned above so that theoutput slit 118 will give a single output beam of light 112 with thecombined wavelengths of light. The output slit 118 replaced the detectorso that no detector will then be used.

Referring to FIG. 12, a more general wavelength Mux/deMux device 120will have multiple input slits 116 and multiple output slits so thatseveral input beams of light 122, each input beam with one or morewavelengths, are dispersed to form several output beams of light 124,each output beam with one or several wavelengths.

Referring to FIG. 13, an integrated version of the wavelength Mux/deMuxdevice 130 will use optical waveguides 132 replacing input and outputslits in which the mouth of each waveguide 132 will be at the locationof the slit it is replacing. As is known to those skilled in the art,the waveguides 132 can be formed by optical fibers or withmultiple-layer dielectric materials with high-refractive-index core toguide optical waves in the form of channel or planar waveguides 132.Such channel or planar waveguides can be formed on a common substrate132.

First, the location of entrance slit or input slit (or waveguide) 502can be adjusted in order to have the best performance for a particulardesign goal. Thus, the location of a entrance slit 502 specified byangle θ₁ with respect to the normal of grating center 504 and thedistance S₁ from grating center 504 is not necessarily on a circle as inthe case for Rowland design mentioned in the prior art.

Second, the location of detector or output slit (or waveguide) can beadjusted in order to have the best performance for a particular designgoal. Thus, the location of detector or output slit (or waveguide) 506,specified by the angle θ₂ with respect to the normal of grating center504 and the distance S₂ from the grating center is not necessarily onthe same circle where entrance slit or input slit (or waveguide) 502 islocated, nor on any other circle.

Third, The relation between θ₁, θ₂, and the initial groove spacing d isgiven by the grating formula,d(sinΘ₂−sinΘ₁)=mλ _(c) /n  (1)

where m is the diffraction order, n is the refractive index of themedium, and λ_(c) is the center of the operation wavelength.

Fourth, the initial groove positions are X₀=(0,0), X₁=(d,R−(R²−d²)^(1/2)) and X⁻¹=(−d, R−(R²−d²)^(1/2)) With these positionvectors, three initial grooves are located on a circle radius R and havethe initial groove spacing of d at the grating center.

Fifth, location of other grooves X_(i)'s are obtained by two conditions.The first of these conditions being that the path-difference betweenadjacent grooves should be an integral multiple of the wavelength in themedium. The first condition can be expressed mathematically by:[d ₁(Θ₁ ,S ₁ ,X ₁)+d ₂(Θ₂ ,S ₂ ,X _(i))]−[d ₁(Θ₁ ,S ₁ ,X _(i−1))+d ₂(Θ₂,S ₂ ,X _(i−1))]=mλ/n,  (2)

(2)

where d₁(Θ₁,S₁,X_(i)) is the distance from a i-th groove located atX_(i) to entrance slit 502 specified by Θ₁ and S₁, d₂ (Θ₂,S₂,X_(i)) isthe distance from i-th groove located X_(i) to detector 502 specified byΘ₂ and S₂, m is the diffraction order, and n is the refractive index ofthe medium. This mathematical expression is numerically exact for theoptical path difference requirement in the diffraction grating and isactively adjusted for every groove on HR-CCG.

The second of these conditions being specific for a particular designgoal of a curved-grating spectrometer. The second condition in generalcan be mathematically expressed asf(Θ₁ ,S ₁,Θ₂ ,X _(i) ,X _(i−1),λ_(c) ,n,m)=const  (3)

Specific examples of the second condition are described later in theapplication. The unknown real variables in both equations (2) and (3)are x- and y-coordinates of the location vector X_(i) of the i-thgroove. For given input-slit location (θ₁, S₁), detector 506 location(θ₂, S₂), and the previous, i.e., (i−1)-th, groove position X_(i−1),X_(i) is completely specified by equations 3 and 4 for a given centerwave-length λ_(c), refractive index n, and the diffraction order m.

The last of the HR-CCG specification ensures that every ray from eachgroove focuses to a single point. This ensures HR-CCG having a largeacceptance angle, and therefore a small spot size.

An exemplary embodiment of HR-CCG specified above is shown in FIG. 6.The radius of curvature at the grating center is R=50 μm. Entrance slitor input slit (or waveguide) 502 is located at an angle Θ₁=55° from thegrating normal and distance S₁=28.68 μm from the grating center.Detector or output slit (or waveguide) 506 is located at an angleΘ₂=27.2° the grating normal and distance S₂=37.65 μm from the gratingcenter. The groove spacing at the grating center is chosen to be d=3.6μm so that diffraction order m=10 is directed toward detector 506located at Θ₂. As shown in FIG. 6, entrance slit 502 and detector 506 isnot located on a circle tangent to the grating center. Three initialgrooves are located at X₀=(0, 0), X₁=(3.6, 0.13), and X⁻¹=(−3.6, 0.13)which form a circle radius R=50 μm. Other groove locations X_(i)'s areobtained with the condition of each groove having a constant angularspacing from entrance slit 502 and optical path-difference condition(Eq. 2). In a mathematical form, this condition is expressed as,

$\begin{matrix}{{\cos\left( {\Delta\;\theta_{i}} \right)} = {\frac{\left( {X_{i} - X_{i\; n}} \right) \cdot \left( {X_{i - 1} - X_{i\; n}} \right)}{{{X_{i} - X_{i\; n}}}{{X_{i - 1} - X_{i\; n}}}} = {const}}} & (4)\end{matrix}$

where X_(in)=(−S₁·sinθ₁, S₁·cosθ1) is the position vector of entranceslit 502, X_(det)=(−S₂·sinθ₂, S₂·cosθ₂) is the position vector ofdetector 506, and Δθ_(i) is the difference in angular position betweensuccessive i^(th) and (i−1)^(th) grooves. In Eq. 4, operator “·” meansthe inner product in vector analysis and defined as A·B≡|A||B| cosθ.Because Δθ_(i) is constant for all grooves, it is same as the angular−position difference between the center groove at X₀ and the firstgroove at X₁, i.e.

${\Delta\;\theta_{1}} = {{arc}\;\cos\frac{\left( {X_{1} - X_{i\; n}} \right) \cdot \left( {X_{0} - X_{i\; n}} \right)}{{{X_{1} - X_{i\; n}}}{{X_{0} - X_{i\; n}}}}}$

In this particular case, the position of entrance slit or input slit (orwaveguide) 502, exit slit or output slit (or waveguide) 506 and theangular spacing between the grooves are X_(in)=(−23.49, 16.45),X_(det)=(−17.26, 33.46), and ΔΘ₁=4.13°. In this example, wave-font ofthe diverging input beam propagating toward the curved grating is slicedinto a set of narrow beams with angular extension ΔΘ by thecurved-grating. Each beam with angular extension ΔΘ undergoes reflectivediffraction by each groove. At a particular wavelength, diffraction at aparticular groove is equivalent to redirecting to a particular narrowbeam into a detector 506 location with Θ₂. The position vectors X_(i)'scalculated from Eq. (2) and Eq. (4) are listed in the Table 2. As shownin FIG. 6, the positions of grooves X_(i) are not on a circle tangent tograting.

X⁻¹⁰ (−23.17, 15.28) X⁻⁹ (−22.24, 12.89) X⁻⁸ (−20.97, 10.60) X⁻⁷(−19.36, 8.43) X⁻⁶ (−17.42, 6.44) X⁻⁵ (−15.17, 4.65) X⁻⁴ (−12.62, 3.10)X⁻³ (−9.80, 1.83) X⁻² (−6.74, 0.87) X⁻¹ (−3.60, 0.14) X₀ (0.00, 0.00) X₁(3.60, 0.14) X₂ (7.30, 0.70) X₃ (11.06, 1.70) X₄ (14.83, 3.13) X₅(18.57, 5.02) X₆ (22.22, 7.36) X₇ (25.73, 10.16) X₈ (29.06, 13.39) X₉(32.16, 17.06) X₁₀ (34.98, 21.15)

The above example has been used for illustration purposes only andshould not be construed in any way as limiting the scope of theinvention.

In an alternative embodiment, the High-Resolution Compact Curved Gratinghas Constant Arc and the Detector or output slit (or waveguide) islocated on a tangent Circle. This embodiment is described below indetail.

In this exemplary embodiment, both entrance slit or input slit (orwaveguide) 502 and detector or output slit (or waveguide) 506 arelocated on a circle tangent to the grating center as in the case ofRowland design mentioned earlier. However, grooves in thiscurved-grating are located such that the arc-length of each groove isthe same. As a result, grooves are not located on a circle nor arespaced with equal distance.

There are two commonly used shapes of grooves in the grating used in thefree-space spectrometer. They are straight line and sinusoidal shape.These two shapes are widely used because of ease of manufacturingprocess. For a curved-grating, ideal shape of reflecting surface not astraight line, but a curved shape of the reflecting surface not astraight line, but a curved shape that can image entrance slit 502 atdetector 506 location. Ideal aberration-free curved mirror is an ellipsewith its focal point located at source and image. Therefore, the idealshape of the groove in a curved-grating is a section of ellipse with itsfocal points at the slit and the detector. In this embodiment, ellipticshape is used for each groove and the length of this elliptic shape ineach groove is kept constant. Center positions of the grooves X_(i)'s inthis example are determined so that the length of each elliptic grooveis the same.

The geometric specification of the HR-CCG with constant arc-length anddetector 506 at a tangent circle is as described below.

First, entrance slit or input slit (or waveguide) 502 is located on acircle tangent to the grating at its center (so-called tangent circle).Therefore, the angle θ₁ and the distance S₁ of entrance slit or inputslit (or waveguide) 502 with respect to the grating center is related byS₁=R cosθ₁, where R is the radius of curvature of the grating center.

First, entrance slit or input slit (or waveguide) 502 is located on acircle tangent to the grating at its center (so-called tangent circle).Therefore, the angle θ₁ and the distance S₁ of entrance slit or inputslit (or waveguide) 502 with respect to the grating center is related byS₁=R cosθ₁, where R is the radius of curvature of the grating center.

Third, the relation between Θ₁, Θ₂, and the initial grove spacing d isgiven by the grating formula, d(sinΘ₂−sinΘ₁)=mλ_(c)/n where m is thediffraction order, n is the refractive index of the medium, and λ_(c) isthe center of the operation wavelength.

Fourth, initial groove positions are X₀ (0,0), X₁=(d, R−(R²−d²)^(1/2))and X⁻¹=(−d, R−(R²−d²)^(1/2)) With these position vectors, three initialgrooves are located on a circle of radius R and have the initial groovespacing of d at the grating center.

Fifth, the location of other grooves X_(i)'s are obtained by thefollowing two conditions. The first condition being the path-differencebetween adjacent grooves should be an integral multiple of thewavelength in the medium, which s mathematically expressed as[d ₁(Θ₁ ,S ₁ ,X _(i))+d ₂(Θ₂ ,S ₂ ,X _(i))]−[Θ₁ ,S ₁ ,X _(i−1))+d ₂(Θ₂,S ₂ ,X _(i−1))]=mλ/n  (2)

The arc-lengths of all the grooves are the same through-out the HR-CCG.This condition can be mathematically expressed as

$\begin{matrix}{{\Delta\; S_{i}} = {{{X_{i} - X_{i\; n}}}\arccos\frac{\left( {\frac{X_{i - 1} + X_{i}}{2} - X_{i\; n}} \right) \cdot \left( {\frac{X_{i} + X_{i + 1}}{2} - X_{i\; n}} \right)}{{{\frac{X_{i - 1} + X_{i}}{2} - X_{i|}}}{{\frac{X_{i} + X_{i + 1}}{2} - X_{i\; n}}}}}} & (5)\end{matrix}$

where ΔS_(i) is the arc-length of i^(th) groove. This equation requiresthe knowledge of X_(i+1), which is still unknown. However, with theconstraint the fact that each X_(i) is located at the center of thegroove, the above expression can be reduced to the following expressionwithout X_(i+1).

$\begin{matrix}{{\Delta\; S_{i}} = {{2{{X_{i} - X_{i\; n}}}{arc}\;\cos\frac{\left( {\frac{X_{i - 1} + X_{i}}{2} - X_{i\; n}} \right) \cdot \left( {X_{i} - X_{i\; n}} \right)}{{{\frac{X_{i - 1} + X_{i}}{2} - X_{i\; n}}}{{X_{i} - X_{i\; n}}}}} = {consi}}} & (6)\end{matrix}$

FIG. 7 shows a specific example of the HR-CCG with constant arc-lengthof the grooves and detector or output slit (or waveguide) 506 at atangent circle. The radius of curvature at the grating center is R=100μm. Entrance slit or input slit (or waveguide) 502 is located at anangle Θ₁=45° from grating normal and a distance S₁=70.71 μm from thegrating center. Detector 506 is located at an angle Θ₂ 37.37° anddistance S₂=79.47 μm from the grating center. Both entrance slit orinput slit (or waveguide) 502 and exit slit or output slit (andwaveguide) 506 are located on a tangent circle of radius 50 μm. Thegroove spacing at the grating center is chosen to be d=4.2 μm so thatdiffraction order m=12 is directed toward detector 506 located at anangle Θ₂ from the grating normal. Three initial grooves are located atX₀=(0, 0), X₁=(4.2, 0.008), and X⁻¹=(−4.2, 0.008) which form a circle ofradius R=100 μm. Other groove locations X_(i)'s are obtained with thecondition of arc-length of each groove ΔS_(i) is the same, i.e. ΔS₁.Equation (2) and (6) are simultaneously solved for a X₁ withX_(in)=(−50, 50), X_(det)=(−48.24, 63.15), and ΔS₁=4.201 μm for a givenX_(i−1). Groove locations, X_(i)'s calculated in this method are listedin Table 3. As shown in FIG. 7, grooves in this grating are not locatedon a tangent circle.

X⁻¹⁵ (−55.43, 23.48) X⁻¹⁴ (−52.90, 20.32) X⁻¹³ (−50.07, 17.38) X⁻¹²(−46.98, 14.68) X⁻¹¹ (−43.67, 12.21) X⁻¹⁰ (−40.17, 9.98) X⁻⁹ (−36.52,7.99) X⁻⁸ (−32.74, 6.24) X⁻⁷ (−28.84, 4.72) X⁻⁶ (−24.86, 3.43) X⁻⁵(−20.81, 2.35) X⁻⁶ (−16.71, 1.48) X⁻⁵ (−20.81, 2.35) X⁻⁴ (−16.71, 1.48)X⁻³ (−12.57, 0.82) X⁻² (−8.39, 0.36) X⁻¹ (−4.20, 0.09) X₀ (0.00, 0.00)X₁ (4.20, 0.09) X₂ (8.39, 0.35) X₃ (12.57, 0.77) X₄ (16.73, 1.34) X₅(20.86, 2.07) X₆ (24.97, 2.94) X₇ (29.04, 3.96) X₈ (33.07, 5.10) X₉(37.06, 6.37) X₁₀ (41.01, 7.76) X₁₁ (44.91, 9.28) X₁₂ (48.77, 10.90) X₁₃(52.57, 12.63) X₁₄ (56.33, 14.47)

The above example has been illustration purposes only and should not inany way limiting the scope of the above-described embodiment orinvention as a whole.

The performance of the HR-CCG with the constant arc-length and detectoron a tangent circle is compared with a Rowland design with the sameparameters such as θ₁, S₁, θ₂, S₂, R, m, d, and λ_(c). It is a directcomparison of a Rowland curved-grating spectrometer described in FIG. 3and a HRC-curved grating spectrometer described in FIG. 7. All theconfiguration parameters are the same for these two spectrometers exceptthe grating itself. Particularly, the imaging properties, that is, howwell entrance slit or input slit (or waveguide) 502 is sharply imaged atthe detector location without aberration are compared. Imagingproperties ultimately determine the resolution of a spectrometer. FiniteDifference Time Domain (FDTD) method is used as a calculation method.FDTD is a Maxwell-equation solver, which evaluates electromagnetic wavewithin a spatial region for a certain period of time. By choosing a finespatial grid size and temporal calculation step, the equation for anelectromagnetic wave and its propagation can be solved with arbitrarilyfine resolution. Imaging properties in these two curved-gratingspectrometers is calculated by propagating a monochromatic light intoentrance slit or input slit (or waveguide) 502 of each spectrometer.FDTD is run until the interference of beams from the entire gratinggroove is completed and forms an image of entrance slit or input slit(or waveguide) 502 at the detector location. The resulting snapshot ofelectric-field near the detector is taken for these two cases as shownin FIG. 8. Entrance slit or input slit (or waveguide) 502 width of 1 μmis used for both simulations and the wavelength of λ=1530, 1550, 1570 nmis used. In FIG. 8A shows the snapshot of electric field at the detectorlocation for the Rowland design described in FIG. 3. As expected, theimage of the entrance slit is blurred due to imperfect grating. For 1 μmentrance slit, the full diffraction angle is about θ_(div)=50° andtherefore, Rowland design breaks down. FIG. 8B shows the snapshot ofelectric field for the HR-CCG with constant arc-length grooves anddetector on a tangent circle. In this case, a sharp aberration freeimage of entrance slit is formed at the detector location. The RS factor(RS=(Δλ/λ)×(L/λ)) in this case is 0.6.

In an alternate embodiment, High-Resolution Compact Curved Grating has aconstant arc with the detector or output slit (or waveguide) beingpresent in-line. With reference to FIG. 7, this embodiment can berealized if the input and exit slits are located along a line such thatS1≈S2. In another alternate embodiment, High-Resolution Compact CurvedGrating has a constant arc with detector or output slit (or waveguide)506 present at an arbitrary location.

In another alternative embodiment, High-Resolution Compact CurvedGrating has a constant angle and detector 506 present on the circle ofradius R, as depicted in FIG. 6. In this embodiment, each groove surfacehas an angular extension (Δθ)_(i) from entrance slit 502. In thisexample, the angular extensions (Δθ)_(I) are kept constant for allgrooves. In addition, both entrance slit 502 and detectors 506 arelocated on a circle of radius R/2, where R is the radius of a circleformed by three initial groove locations X0, X1, and X2.

In as yet another alternate embodiment, High-Resolution Compact CurvedGrating has a constant angle and detector or output slit (or waveguide)506 is present on the circle of radius R/2, as depicted in FIG. 6. Inthis embodiment, each groove surface has an angular extension (Δθ)_(i)from entrance slit or input slit (or waveguide) 502. In this example,the angular extensions (Δθ)_(i) are kept constant for all grooves. Inaddition, both entrance slit or input slit (or waveguide) 502 anddetectors or output slits (or waveguides) 506 are located on or near acircle of radius R/2, where R is the radius of a circle formed by threeinitial groove locations X0, X1, and X2.

In as yet another alternate embodiment, High-Resolution Compact CurvedGrating has a constant angle with detector or output slit (or waveguide)506 present at an arbitrary location.

In as yet another alternate embodiment, High-Resolution Compact CurvedGrating has groves lying on or near the circle of radius R (thenear-Rowland Case) where R is the radius of a circle formed by threeinitial groove locations X0, X1, and X2, and the detector or output slit(or waveguide) 506 is present on the circle of radius R/2, as depictedin FIG. 14. In this embodiment, each groove surface has an angularextension (Δθ)_(i) from entrance slit or input slit (or waveguide). Inthis example, the angular extensions (Δθ)_(i) are chosen so that eachgroove lies on or near the circle of radius R. More specifically, thelocation of other grooves X_(i)'s are obtained by the following twoconditions. The first condition being the path-difference betweenadjacent grooves should be an integral multiple of the wavelength in themedium, which is mathematically expressed as[d ₁(θ₁ ,S ₁ ,X _(i))+d ₂(θ₂ ,S ₂ ,X _(i))]−[d ₁(θ₁ ,S ₁ ,X _(i−1))+d₂(θ₂ ,S ₂ ,X _(i−1))]=mλ/n,  (7)

Secondly, the angular locations of the grooves are chosen so that eachgroove is located at or near the circle of radius R throughout theHR-CCG, where R is the radius of a circle formed by three initial groovelocations X0, X1, and X2.

In another alternate embodiment, the High-Resolution Compact CurvedGrating with groves on or near the circle of radius R (the near-Rowlandcase) has detector or output slit (or waveguide) present at an arbitrarylocation.

While preferred embodiments of the invention have been illustrated anddescribed, it will be clear that the invention is not limited to theseembodiments only. Numerous modifications, changes, variations,substitutions and equivalents will be apparent to those skilled in theart without departing from the spirit and scope of the invention asdescribed in the claims.

1. A wavelength multiplexer/demultiplexer suitable for analyzing aspectra composition of an optical beam, the wavelengthmultiplexer/demultiplexer enabling a detection of light of a particularwavelength, the wavelength multiplexer/demultiplexer comprising: aninput slit for allowing an entry of the optical beam into aspectrometer, a location of the input slit being adjustable forcontrolling a performance of the wavelength multiplexer/demultiplexer;an output slit for allowing the exiting of the optical beam, a locationof the output slit being adjustable for controlling the performance ofthe wavelength multiplexer/demultiplexer; and a curved grating foranalyzing the spectra composition of the optical beam, the curvedgrating comprising a plurality of grooves, a distance between thegrooves being dependent on the location of the input slit and the outputslit, a center of operation wavelength, the diffraction order, arefractive index of the medium and on the location of the adjacentgrooves, such that a path difference between two adjacent grooves is anintegral of the center of operation wavelength, wherein an arc length ofthe grooves is the same.
 2. The wavelength multiplexer/demultiplexer asrecited in claim 1, wherein the input slit and the output slit arelocated on a tangent circle.
 3. The wavelength multiplexer/demultiplexeras recited in claim 1, wherein the curved grating has one of a straight,sinusoidal and elliptical shapes.
 4. The wavelengthmultiplexer/demultiplexer as recited in claim 1, wherein thespectrometer is in accordance with a Littrow configuration.
 5. Thewavelength multiplexer/demultiplexer as recited in claim 1, wherein thespectrometer is used as a wavelength dispersion element in a photonicintegrated circuit.
 6. The wavelength multiplexer/demultiplexer asrecited in claim 1, wherein the wavelength multiplexer/demultiplexer isan isolated optical wavelength multiplexer/demultiplexer using discretecomponents, the discrete components including slits, gratings,spectrometer casing detector, detector array and motor drive.
 7. Thewavelength multiplexer/demultiplexer as recited in claim 1, wherein thewavelength multiplexer/demultiplexer is an integrated optical wavelengthmultiplexer/demultiplexer using an optical waveguide as the input slitand an optical waveguide as the output slit.
 8. The wavelengthmultiplexer/demultiplexer as recited in claim 7, wherein the opticalwaveguides and the grating are fabricated on a single substrate.
 9. Thewavelength multiplexer/demultiplexer as recited in claim 1, wherein anarc length of each of the grooves is the same.
 10. The wavelengthmultiplexer/demultiplexer as recited in claim 1, wherein an angularspacing of each of the grooves is the same.
 11. The wavelengthmultiplexer/demultiplexer as recited in claim 1, wherein the grooves areat or near a circle of radius R where R is the radius of a circle formedby three initial groove locations.
 12. A compact curved grating suitablefor analyzing the spectra composition of an optical beam, the opticalbeam being incident on the compact curved grating via an input slit, theanalyzed optical beam from the compact curved grating being incident onan output slit, the compact curved grating comprising a plurality ofgrooves, the distance between the grooves being dependent on thelocation of the input slit and the output slit, the center operationwavelength, a diffraction order, the refractive index of the medium andon the location of the adjacent grooves, such that a path differencebetween two adjacent grooves is an integral of the center of operationwavelength, wherein an arc length of the grooves is the same.
 13. Amethod for analyzing a spectra composition of an optical beam, themethod comprising: adjusting a location of an input slit in order tohave best performance at a particular design goal, the optical beamentering the wavelength multiplexer/demultiplexer through the inputslit; adjusting a location of an output slit in order to have bestperformance at a particular design goal, the spectra composition of theoptical beam being separated by the output slit; and using a compactcurved grating in order to analyze the spectra composition of theoptical beam, the compact curved grating comprising a plurality ofgrooves, the step of using the compact curved grating further comprisingthe step of: calculating initial groove spacing using the informationrelating to location of the input slit and the output slit, center ofthe operation wavelength, refractive index of the medium and thediffraction order; and determining the positions of other grooves, thepositions being determined by ensuring that path difference betweenadjacent grooves is an integral multiple of the wavelength in themedium, according to the following mathematical expression:[d ₁(Θ₁ ,S ₁ ,X _(i))+d ₂(Θ₂ ,S ₂ ,X _(i))]−[d ₁(Θ₁ ,S ₁ ,X _(i−1))+d₂(Θ₂ ,S ₂ ,X _(i−1))]=mλ/n wherein d₁(Θ₁,S₁,X_(i)) is a distance fromone of the plurality of grooves located at X_(i) from the entrance slit,d₂(Θ₂,S₂,X_(i)) is a distance from one of the plurality of grooves formthe detector, m is a diffraction order and n is a refractive index ofthe medium, and further wherein an arc length of each of the grooves isthe same; and adjusting the initial groove spacing and the positions ofother grooves according to the calculating and determining steps.